Modification of k-ε Turbulent Model Using Kinetic Energy Preserving Method

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1 Numerical Heat Transfer, Part B: Fundamentals An Internatinal Jurnal f Cmutatin and Methdlgy ISSN: (Print) (Online) Jurnal hmeage: htt:// Mdificatin f k-ε Turbulent Mdel Using Kinetic Energy Preserving Methd Ali Javadi, Mahmud Pasandideh-Fard & Majid Malek-Jafarian T cite this article: Ali Javadi, Mahmud Pasandideh-Fard & Majid Malek-Jafarian (015) Mdificatin f k-ε Turbulent Mdel Using Kinetic Energy Preserving Methd, Numerical Heat Transfer, Part B: Fundamentals, 68:6, T link t this article: htt://dx.di.rg/ / Published nline: 01 Oct 015. Submit yur article t this jurnal Article views: View related articles View Crssmark data Full Terms & Cnditins f access and use can be fund at htt:// Dwnlad by: [Mahmud Pasandideh Fard] Date: 11 December 015, At: 08:0

2 Numerical Heat Transfer, Part B, 68: , 015 Cyright # Taylr & Francis Gru, LLC ISSN: rint/ nline DOI: / MODIFICATION OF k-ɛ TURBULENT MODEL USING KINETIC ENERGY PRESERVING METHOD 1. INTRODUCTION Ali Javadi 1, Mahmud Pasandideh-Fard 1, and Majid Malek-Jafarian 1 Deartment f Mechanical Engineering, Ferdwsi University f Mashhad, Mashhad, Iran Deartment f Mechanical Engineering, University f Birjand, Birjand, Iran In this article, the kinetic energy reserving (KEP) scheme and, als, the way f alying this scheme t the k ε turbulent mdel are intrduced. This study aims t intrduce a stable methd in which a few artificial dissiatin terms are added t the gverning equatins in a way that the intensity f the slutin fluctuatins is reduced and, therefre, the rblem stability increases. In accrd with the imrtance f the study f turbulent flws, the effects f the fluctuating velcity terms n the calculatin f all fluxes in the gverning equatins are scrutinized as well. Als, the influence frm alying the KEP scheme n the k ε turbulent mdel is investigated. This article reveals that by using the KEP scheme and, afterwards, imrving the discretizatin methd f the velcity fluctuatin terms in k ε equatins, the accuracy f the results btained is enhanced withut a need t add artificial dissiatin terms (r by minimizing their values). This article ursues the idea, dating back t the early days f scientific cmuting, f the energy methd fr stability f numerical analysis. Very ften, artificial dissiatin terms are added t the equatins t cature shck waves and increase stabilities. Fr examle, in the scalar methd rsed by Jamesn, Smith, and Turkel [1, ] (JST methd), in regins where there is a strng ressure gradient, a cnsiderable amunt f artificial dissiatin term is added. If artificial dissiatin terms are nt cmletely cntrlled, daming can ccur in all ints f the cmutatinal dmain. In such a situatin, sme flw rerties such as turbulent vrtexes may nt be catured by the numerical simulatin. In rder t reslve the rblems mentined, Jamesn [, ] has recently develed a new scheme in which the ttal kinetic energy is reserved. By reserving the ttal kinetic energy, this scheme rvides the stability needed fr the calculatin f the viscus flw even in cmlicated rblems including shck waves. Since n artificial dissiatin terms are added t the equatins (r their values are negligible), sme nticeable rerties are exected t be catured in this scheme that d nt exist in the ther similar schemes []. Received 5 February 015; acceted 19 May 015. Address crresndence t Mahmud Pasandideh-Fard, Ferdwsi University f Mashhad, P.O. Bx , Azadi Sq., Mashhad, Khrasan Razavi, Iran. fard_m@um.ac.ir 55

3 MODIFICATION OF k Ɛ TURBULENT MODEL 555 NOMENCLATURE A area f the surface flux, m C D drag cefficient C L lift cefficient C ressure cefficient C secific heat caacity, J=kg: k ds surface element, m dv vlume element, m E ttal energy, J=kg f flux vectr H ttal enthaly, J=kg k secific kinetic energy, m =s k fluctuating kinetic energy, m =s k thermal cnductivity, W=m: k M Mach number n j nrmal surface vectr, m ressure, Pa Pr Prandtl number q heat flux, w Re Reynlds number S surface between and, m t time, s T temerature, K u velcity, m=s u, v, w velcity cmnents in x, y, z directins, resectively, m=s vl element vlume, m w state vectr x crdinate directin, m d Krneker delta e dissiatin rate, m =s k dynamic viscsity, Pa s m dynamic viscsity, Pa s t kinematic viscsity, m =s q density, kg=m r viscus stress ðkg=m:s Þ Subscrits b m k, m j, m i k, j, i bundary nde revius nde in i, j, and k directins mean nde neighbr nde next nde in i, j, and k directins Suerscrits i, j, k cunting index (indicating the crdinate directin) n number f time ste mean value 0 fluctuating term Previus wrks by the authrs shw that the kinetic energy reserving (KEP) methd leads t the same results as the ther methds until the number f grids is lw [5]. The challenge is t imrve the KEP methd t gain better results at lw ints f the grid. T d this, sme crrectins have t be made in the KEP methd. After making such crrectins, sme inaccuracies may ccur in the kinetic energy reservatin. Hwever, due t the imrvement f the results fr a lw number f the grids, such inaccuracies can be neglected. Bundary cnditins are f great imrtance in this methd. Wang Li et al. [6] have studied a secnd-rder additinal surce term methd fr handling bundary cnditins. Crrected bundary cnditins must be cnsidered in this scheme. There are several research rerts abut this methd. The KEP scheme has been studied in ne-dimensinal viscus flw in a shck tube by Jamesn [7], and in tw-dimensinal viscus flw in a shck tube and als tw-dimensinal viscus flw arund a lunging airfil by Allaneu and Jamesn [8, 9]. Lehmkuhl et al. [10] used the KEP methd fr the flw arund wind turbine blades. Trias et al. [11] resented a fully -cnservative discretizatin f the Navier-Stkes equatins fr unstructured meshes. This methd was tested fr a buyancy-driven turbulent flw. Baez Vidal et al. [1] cmared the KEP and Gdunv schemes n the flw arund a NACA 001 airfil. Herbin and Latche [1] resented a kinetic energy reserving eratr fr the MAC discretizatin f cmressible Navier-Stkes equatins. Edh and Karagzian [1] alied KEP discretizatin schemes fr

4 556 A. JAVADI ET AL. high-reynlds-number rulsive alicatin. Mrever, Chandrashekar [15, 16], Kk [17], and Yan and Jin [18] scrutinized this methd in their articles. Als, Javadi and Pasandideh-Fard [5] have alied the KEP methd in a nedimensinal inviscid cnvergent-divergent nuzzle flw, tw-dimensinal inviscid flw ver a bum, and tw-dimensinal viscus flw ver a NACA 001 airfil. It was shwn that the KEP scheme is mre accurate if the number f mesh ints is increased; and, in cntrast t ther schemes, there is n limit in increasing the grid ints. Studying the kinetic energy f the flw is f great imrtance, esecially in study f the turbulence existing in the flw. Mst flws are naturally turbulent. Therefre, the analysis f the effect f the turbulence is very imrtant. There are several methds t study the effects f the turbulence. Fr examle, the RANS, LES, and DNS methds can be nted. Direct numerical simulatin (DNS) f the turbulence is the mst crrect arach t slve turbulent flws. Large-eddy simulatin (LES) is similar t DNS, because bth ffer three-dimensinal and time-deendent slutin fr the Navier-Stkes equatins [19]. S they bth need a fine enugh grid. At high Reynlds numbers, the DNS methd can be relaced by the LES methd. Althugh LES has been cmutatinally develed mre than DNS and therefre is a gd alternative t DNS at higher Reynlds number, its alicatin n engineering flws remains exensive, unless using wall functins. Accrding t the existing rblems in these methds, it is useful t aly Reynlds-averaged Navier-Stkes (RANS) mdels t reduce the cmutatinal csts [0]. Simle turbulent mdels (such as zer-equatin mdels t tw-equatin mdels) create almst accurate results fr simle flws in which nly ne cmnent f the Reynlds stresses in the mmentum equatins is imrtant [1]. One f the useful RANS mdels is the k ε turbulent mdel. Several cases have been mdeled by this turbulent mdel. Fr examle, Maa and Zhanga [], Pulat et al. [], and Chi et al. [] used this mdel t study the behavir f the kinetic energy f a fluid. The KEP methd is a new scheme which has been less studied s far. Mrever, utilizatin f this methd in different tyes f flws has nt been erfrmed yet. Jamesn has derived the flux equatins nly fr the main velcity u. The searatin f the main velcity int averaged and fluctuated velcity cmnents creates a new frmula fr calculating the amunt f u i u j r u i u j. The influence f imlementing these relatins n different mdels and methds causes the calculatin f the fluctuating terms t change. In fact, the mst imrtant innvatins f this article are t imlement the relatins f the fluctuating terms and, als, t make changes in calculatin f u i u j in the k ε mdel. In the fllwing, the KEP methd and its alicatin n the k ε turbulent mdel are intrduced.. INTRODUCTION OF THE KINETIC ENERGY PRESERVING (KEP) METHOD This sectin intrduces the kinetic energy-reserving scheme. It shuld be nted that the equatins given in Sectin.1 are derived frm [5]. Als, the relatins resented in Sectins. and. are btained fr the first time by the authrs. The mathematical details f these tw sets f equatins are cmletely given in Aendixes A and B.

5 MODIFICATION OF k Ɛ TURBULENT MODEL The KEP Methd and Its Imlementatin in the Navier-Stkes Equatins (t Calculate the Existing Flux) The gverning equatins, including the Navier-Stkes (N-S) and energy equatins, are as fllws: where qw qt þ q qx i f i ðwþ ¼ 0 q qv i qu 1 qu i u 1 r i1 þ d i1 w ¼ 6 qu 7 qu 5 f i ¼ qu i u r i þ d i 6 7 qu i u r i þ d i 5 qe qu i H u j r ij q j Als, the viscus stress tensr and the heat flux are btained frm the relatins () and (): r ij ¼ m þ qui þ quj qx i kd ij qu k qx k q j ¼k qt In the abve equatins, m and k are the viscsity cefficients, in which k is usually defined as k ¼ m: dij is the Krnecker delta and k is the thermal cnductivity. The k variable is defined as the kinetic energy, as can be seen in Eq. (5): k ¼ q ui qk qw ¼ " # ui ; u1 ; u ; u ; 0 By derivatin f k with resect t time, Eq. (6) is btained: qk qt ¼ q 1 ¼ u i q qt qui qu i u i qq qt qt By using the equatin f state and the flux vectrs, if the satial derivatives are relaced with time derivatives, Eq. (7) is btained: "! # qk qt þ q u j þ q vi u i r ij ¼ qvj qui rij qxj ð1þ ðþ ðþ ðþ ð5þ ð6þ ð7þ By integrating the abve equatin, Eq. (8) is btained: Z Z "! # Z q kdv ¼ u j þ q vi u i r ij n j ds þ quj qui rij qt qxj dv X qx X ð8þ

6 558 A. JAVADI ET AL. where n j is the nrmal surface vectr and ds and dv are the surface and vlume elements, resectively. Equatin (8) shws the ttal kinetic energy reserving equatin fr three-dimensinal viscus flw. Nw the equatins are discretized by the finite-vlume methd. Fr any nde, the neighbr ndes are secified by. Als, the surface between these tw ndes is defined as A. n i is reresentative f the nrmal surface vectr, in which i dentes the directin f this vectr. The fllwing equatins can be written fr a cntrl vlume: r n i ¼ni S i ¼ A n i S i ¼X S i The cnservatin equatin cmes in the frm belw: vl qw qt vl qw qt þ X neighbr f i :ni A ¼ 0 þ X neighbr f i :Si ¼ 0 where the state vectr and the flux vectr can be defined as ðqv q i Þ q u 1 qu i u 1 þ di1 r i1 w ¼ 6 q u 7 q u 5 f i ¼ qu i u þ di r i qu i u q E þ 6 di r i 7 5 ðqu i HÞ þðu j r ij þ q j It can be shwn that if the relatins (15) and (16) are true, qu i u j ¼ 1 qui uj þ uj d ij r ij ¼ 1 dij r ij þ 1 dij r ij Equatin (1) cmes in the frm d dt X vl k ¼ X b Sj b u j u i b b b þ q b þ X X! u i þ ui S i u i b rij b rij! X Þ u i þ ui S i ð9þ ð10þ ð11þ ð1þ ð1þ ð1þ ð15þ ð16þ! ð17þ That is the discrete frm f Eq. (8), where b reresents the bundary ints. Fr a variable q, the averaged value is defined as q ¼ q þ q ð18þ

7 MODIFICATION OF k Ɛ TURBULENT MODEL 559 S qu i u j ¼ qui uj ð19þ Different tyes f calculatin f the flux can be determined t ensure the reservatin f ttal kinetic energy. Fr examle, qu i ¼ q u i qu i ¼ qui ð0þ ð1þ Similarly, qu i u j ¼ q u i uj qu i H ¼ q u i H.. Alying the KEP Methd n Fluxes Including Velcity Fluctuatins It can be shwn that if the fllwing relatins are true fr the fluctuating velcity terms, the ttal kinetic energy in the cmutatinal dmain is reserved. These relatins are fully rved in Aendix A. qu i u j ¼ 1 qui uj þ uj qu i u j ¼ 1 qui qu i u j ¼ 1 qui qu i u j ¼ 1 qui uj þ uj uj þ uj uj þ uj By searating the average and fluctuating cmnents f the velcity, Eqs. (8) and (9) are btained t calculate the viscus fluxes f the mmentum equatin: ¼ 1 ¼ 1 þ 1 þ 1 ðþ ðþ ðþ ð5þ ð6þ ð7þ ð8þ ð9þ

8 560 A. JAVADI ET AL... Alying the KEP Methd n the Fluxes f the k and ε Equatins The k and ε equatin are defined by Eqs. (0) and (1): q qk qt þ qk uj ¼ q u j þ q u ui j þ l q qk þ qu i u j qu i qe ð0þ q ðqeþþqu j q qt ðþm e q " qe þ q e qui q # q ¼ n k qx k qx k uj qxi qx k qu i qu j quj qu i qu i qx k m qxk qx k qx k m q u i qu i qu j qu i qx k uj mqui qxk qx k qx k The first and the third terms f the right-hand side f Eq. (0) and all f the terms n the right-hand side f Eq. (1) are btained by mdeling. Next, Eqs. () (5) illustrate hw t imlement KEP methd in the k ε equatin in rder t calculate the flux. It is wrth nting that, as can be seen in the fllwing relatins, the average value f the flux is nt calculated. In ther wrds, the significance f the main ints in btaining the results is mre than the neighbr ints. These relatins are rved in Aendix B. qu i k ¼ 1 qui k þ k m qk ¼ m qk þ 1 m qk qu i e ¼ 1 qui e þ e m qe ¼ m qe þ 1 m qe ð1þ ðþ ðþ ðþ ð5þ Figure 1. Gemetry and mesh generatin f the flat late.

9 MODIFICATION OF k Ɛ TURBULENT MODEL 561. RESULTS AND DISCUSSION Figure. Cnvergence histry, Re ¼ In the fllwing, tw-dimensinal viscus flw ver a flat late is first investigated, and then the results f the high-reynlds transnic NACA 001 airfil achieved by using this methd are resented..1. Tw-Dimensinal Viscus Flw ver a Flat Plate In this sectin, the results btained frm alying the KEP methd in the k ε turbulent mdel are resented and cmared with k ε turbulent mdel results when the KEP methd is nt added t them. Figure. Velcity rfile, X ¼ 0.8, Re ¼ 10 6.

10 56 A. JAVADI ET AL. Figure. Distributin f frictin cefficient, Re ¼ The gemetry cnsidered in this study is a flat late. In this sectin, the influence f the gverning arameters n the results is investigated fr the late crss sectin at x ¼ 0.8. All f the arameters are dimensinless. The Mach number and Reynlds number f the air flw are assumed t be 0.6 and 10 6, resectively. Mrever, the secific heat caacity is C P ¼ 1,00 J=kg. k, and the Prandtl number is Pr ¼ 0.7. A schematic f the rblem is shwn in Figure 1 alng with the mesh generatin. At the inlet, the amunts f all arameters are assumed t be equal with thse f the free stream. Values f velcity, density, k, and ε in the exit crss sectin are cnsidered t be the same as their values at the ints next t them in the dmain, Figure 5. Velcity rfile fr KEP methd with different artificial dissiatin, X ¼ 0.8, Re ¼ 10 6.

11 MODIFICATION OF k Ɛ TURBULENT MODEL 56 Figure 6. Distributin f frictin cefficient fr KEP methd with different artificial dissiatin, Re ¼ and ressure values in the exit crss sectin are equal t the ressure f the free stream. Fr the ints at the bttm, values f density and ressure are set t be equal t their uer ints. Velcity, k, and ε ver the flat late are cnsidered t be zer. The generated mesh t study the results is (60 0) ints. The grid generatin is unifrm in the x directin, but an exansin cefficient f 1. is alied in the y directin. Figure shws the cnvergence histry fr bth methds. The cnvergence rcess ccurs faster fr the KEP methd. Figure shws the velcity rfile in crss sectin x ¼ 0.8. As can be seen, the results achieved by using the tw methds Figure 7. Cnvergence histry fr KEP methd with different artificial dissiatin, Re ¼ 10 6.

12 56 A. JAVADI ET AL. Figure 8. Mesh generatin fr NACA 001 airfil. are clse tgether, hwever, a slight imrvement in the results btained using the KEP can be bserved. Figure shws the values f lcal frictin cefficient. By fcusing n this figure, it can be nticed that with alying the KEP methd in the k ε mdel, the results are imrved. Using the KEP methd in the k ε mdel has anther significant asset, which is exressed in this sectin. As reviusly mentined, a few artificial dissiatin terms are added t the gverning equatins withut making the numerical slutin diverge r reducing the accuracy f the results. This sectin resents the results f alying the KEP methd, while in this case a little artificial dissiatin term is added t Figure 9. Cefficient ressure versus X (M ¼ 0.7, a ¼ 1.9).

13 MODIFICATION OF k Ɛ TURBULENT MODEL 565 Figure 10. Cefficient ressure versus X (M ¼ 0.55, a ¼ 8.). the equatin. The adding rcedure f the dissiatin terms is accmlished in three stes. In the first ste, the dissiatin term, which is 0% f that f the SCDS methd [1], is added t the gverning equatins. Next, 5% f the dissiatin term f the SCDS methd is added and, finally, the KEP methd is used withut any artificial dissiatin term. Figures 5 7, illustrate the results achieved. As catured in these figures, it is ssible t make the simulatin cnverge, withut rducing large fluctuatins, by adding nly a few artificial dissiatin terms. In such a henmenn, the accuracy f the results is nt reduced. On the ther hand, withut using the KEP methd, when the artificial dissiatin terms are eliminated r their values decrease Figure 11. Cefficient ressure versus X (M ¼ 0.799, a ¼.6).

14 566 A. JAVADI ET AL. Figure 1. Mach cnturs (M ¼ 0.7, a ¼ 1.9).

15 MODIFICATION OF k Ɛ TURBULENT MODEL 567 t 5% r 0% f thse f the SCDS methd, the numerical simulatin diverge. Cnsequently, the results resented are related nly t the situatin when the KEP methd is used... Lw-Dissiatin, High-Reynlds Transnic NACA 001 Airfil The secnd case is a NACA 001 airfil. A series f general C meshes generated with a hyerblic slver are used fr the Rw field cmutatins. The ttal number f grid ints is (98 99). Figure 8 illustrates a view f the mesh which is used fr the NACA 001 cases. Fr all these cases, the Reynlds number is 9 millin. The KEP methd with lw dissiatin (0% f the artificial dissiatin f the SCDS methd) is cnsidered and als cmared with the exerimental data f Harris [6] and cmutatinal data f Maksymiuk [7]. First, cmutatins were carried ut fr the NACA 001 airfil at three secified cnditins. Mach numbers, angles f attack, and Reynlds numbers are secified, and the variatins f C with resect t dimensinless dislacement X are ltted. The KEP results are cmared with the exerimental data f Harris [6] in Figures The results agree very well with exeriment. Mach cnturs f the first case (M ¼ 0.7, a ¼ 1.9) are shwn in Figure 1. In Figures 1 and 1, the variatin f lift cefficient with angle f attack and als with drag cefficient at a free-stream Mach number f 0.7 is illustrated. The results which were cmuted agree well with exerimental data at gemetric angles f attack u t abut 5.0. The drag cefficient (Figure 1) is als calculated reasnably well in cmarisn t exerimental data. Figure 15 shws the drag cefficient distributin with resect t Mach number. Results are cmared with Maksymiuk s results [7]. Hwever, the KEP results are in agreement with the cmutatinal data f Maksymiuk. Figure 1. Lift cefficient versus angle f attack (M ¼ 0.7).

16 568 A. JAVADI ET AL. Figure 1. Lift cefficient versus drag cefficient (M ¼ 0.7). Figure 15. Drag cefficient versus Mach number (a ¼ 0.0).. CONCLUSION In this article, the kinetic energy reserving scheme, and the way f alying this scheme t the k ε turbulent mdel, were intrduced. The results btained shwed that by imlementing the KEP methd, the accuracy f the results is imrved. Furthermre, by using this scheme, it is ssible t attain a cnverged slutin and, als, accurate results withut a need t add any artificial dissiatin terms (r by minimizing their values). Obtaining an accurate slutin is als ssible when nly a few dissiatin terms are added. If the KEP methd is nt imlemented r the term f the artificial dissiatin is eliminated r reduced, the numerical simulatin diverges.

17 FUNDING MODIFICATION OF k Ɛ TURBULENT MODEL 569 The authrs gratefully acknwledge the surt f the Deartment f Mechanical Engineering f Ferdwsi University f Mashhad. REFERENCES 1. A. Jamesn, W. Schmidt, and E. Turkel, Numerical Slutins f the Euler Equatins by Finite Vlume Methds Using Runge-Kutta Time-Steing Schemes, AIAA J., , R. Swansn and E. Turkel, Artificial and Central Difference Schemes fr the Euler and Navier Stkes Equatins, AIAA 8th Cmutatins Fluid Dynamics Cnference, New Yrk, , A. Jamesn and Y. Allaneu, Kinetic Energy Cnservatin Discntinuus Galerkin Scheme, 9th Aersace Sciences Meeting by the AIAA, Flrida, January 7, A. Jamesn, Frmulatin f Kinetic Energy Preserving Cnservative Schemes fr Gas Dynamics and Direct Numerical Simulatin f One-Dimensinal Viscus Cmressible Flw in a Shck Tube Using Entry and Kinetic Energy Preserving Schemes, AIAA J., , A. Javadi and M. Pasandideh-Fard, Analysis f One and Tw Dimensinal Inviscid and Tw Dimensinal Viscus Flws Using Kinetic Energy Preserving Methd, Iran. J. Mech. Eng., Trans. ISME, vl. 1, , W. Li, B. Yu, X. Wang, P. Wang, and W. Ta, Study n the Secnd-Order Additinal Surce Term Methd fr Handling Bundary Cnditins, Numer. Heat Transfer B, vl. 6,. 61, A. Jamesn, Energy Estimates fr Nnlinear Cnservatin Law with Alicatins t Slutins f the Burgurs Equatin and One-Dimensinal Viscus Flw in a Shck Tube by Central Difference Schemes, 18th Cmutatinal Fluid Dynamics Cnference by the AIAA, Miami, June 8, A. Jamesn and Y. Allaneu, Direct Numerical Simulatins f a Tw Dimensinal Viscus Flw in a Shck Tube Using a Kinetic Energy Preserving Schemes, 19th Cmutatinal Fluid Dynamics Cnference by the AIAA, Texas, June 5, A. Jamesn and Y. Allaneu, Direct Numerical Simulatins f Plunging Airfils, 8th Aersace Sciences Meeting by the AIAA, Flrida, January,., O. Lehmkuhl, A. Vidal, D. Perez-Segarra, and A. Oliva, A Filtered Kinetic Energy Preserving Finite Vlumes Scheme fr Cmressible Flws, BME Deartment f Fluid Mechanics, Cnference n Mdeling Fluid Flw (CMFF 1), F. X. Trias, O. Lehmkuhl, A. Oliva, and C. D. Perez-Segarta, Symmetry-Preserving Discretizatin f Navier-Stkes Equatins n Cllcated Unstructured Grids, J. Cmut. Phys., vl. 58,. 6 67, A. Baez Vidal, J. B. Pedr, O. Lehmkuhl, I. Rdriguez, and C. D. Perez-Segarta, Cmaring Kinetic Energy Preserving and Gdunv Schemes n the Flw arund a NACA 001, 6th Eurean Cnference n Cmutatinal Fluid Dynamics, R. Herbin and J. C. Latche, A Kinetic Energy Preserving Cnvectin Oeratr fr the MAC Discretizatin f Cmressible Navier-Stkes Equatin, Math. Mdel. Numer. Anal., hal , versin 1, A. Edh and A. Karagzian, Kinetic Energy-Preserving Discretizatin Schemes fr High Reynlds Number Prulsive Alicatin, 66th Annual Meeting f the APS Divisin f Fluid Dynamics, vl. 58, n. 18, P. Chandrashekar, Kinetic Energy Preserving and Entry Stable Finite Vlume Schemes fr Cmressible Euler and Navier-Stkes Equatins, Cite as: arxiv: [cs.na], 01.

18 570 A. JAVADI ET AL. 16. D. Ray and P. Chandrashekar, Entry Stable Schemes fr Cmressible Euler Equatins, Int. J. Numer. Anal. Mdel., vl.,. 5 5, J. C. Kk, A Symmetry and Disersin-Relatin Preserving High-Order Scheme fr Aer Acustics and Aerdynamics, Eurean Cnference n Cmutatinal Fluid Dynamics, B. Yan and S. Jin, A Successive Penalty-Based Asymttic-Preserving Scheme fr Kinetic Equatin, SIAM J. Sci. Cmut., vl. 5, n.1, M. German, U. Pimelli, P. Min, and W. H. Cabt, A Dynamic Sub Grid-Scale Eddy Viscsity Mdel, Phys. Fluids, vl. A, , D. Vanden-Abeele, D. Snyder, Y. Detundt, and G. Degrez, A Kinetic Energy Preserving P1 Is P=P1 Finite-Element Methd fr Cmuting Unsteady Incmressible Flws, Cmut. Fluid Dynam.,. 67 7, S. V. Patankar and V. Suhas, Numerical Heat Transfer and Fluid Flw, Hemishere, Washingtn, DC, Y. Maa and Y. Zhanga, Evaluatin f Turbulent Mdels fr Natural Cnvectin f Cmressible Air in a Tall Cavity, Numer. Heat Transfer B, vl. 6,. 51 6, 01.. E. Pulat, M. K. Isman, A. B. Etemglu, and M. Can, Effect f Turbulence Mdels and Near-Wall Mdeling Araches n Numerical Results in Imingement Heat Transfer, Numer. Heat Transfer B, vl. 60, , S. K. Chi, E. K. Kim, and S. O. Kim, Cmutatin f Turbulent Natural Cnvectin in a Rectangular Cavity with the k e v f Mdel, Numer. Heat Transfer B, vl. 8, , A. Jamesn, The Cnstructin f Discretely Cnservative Finite Vlume Schemes That Als Glbally Cnserve Energy r Entry, Rert ACL, , C. D. Harris, Tw-Dimensinal Aerdynamic Characteristics f the NACA 001 Airfil in the Lngley 8-Ft Transnic Pressure Tunnel, NASA TM-8197, C. M. Maksymiuk and T. H. Pullian, Viscus Transnic Airfil Wrksh Results Using ARCD, 5th Aersace Sciences Meeting by the AIAA, Nevada, January APPENDIX A. EQUATIONS INCLUDING FLUCTUATING TERMS IN THE KEP METHOD FOR THREE-DIMENSIONAL VISCOUS FLOW Necessary cnditins fr calculating the flux, in the KEP methd fr threedimensinal viscus flws, are exressed in Eqs. (15) and (16). The fllwing equatins can be written t calculate the flux: ðquu ðquv Þ ¼ 1 ðquþ u þ u Þ ¼ 1 ðquþ v þ v ðquwþ ¼ 1 ðquþ w þ w ðqvu Þ ¼ 1 ðqvþ u þ u ðqvvþ ¼ 1 ðqvþ v þ v ðqvwþ ¼ 1 ðqvþ w þ w ð6-aþ ð6-bþ ð6-cþ ð6-dþ ð6-eþ ð6-fþ

19 MODIFICATION OF k Ɛ TURBULENT MODEL 571 ðqwuþ ¼ 1 ðqwþ u þ u ðqwvþ ¼ 1 ðqwþ v þ v ðqwwþ ¼ 1 ðqwþ w þ w ð6-gþ ð6-hþ ð6-iþ Fr examle, Eq. (6-B) is checked. Fr ther equatins we can act similarly. By relacing mean and fluctuating velcity terms in Eq. (6-B), the fllwing exressin is btained: 1 h i ðqu Þ v þ ðqu Þ v þ ðqu Þ v þ ðqu Þ v þ 1 h i ðqu Þ v þ ðqu Þ v þ ðqu Þ v þ ðqu Þ v ð7þ h i ðquv Þ þðquv Þ þðquv Þ þðquv ¼ 0 Using the time average frm Eq. (7), 1 h i ðqu Þ v þ 0 þ ðqu Þ v þ 0 þ 1 h i 0 þ ðqu Þ v þ 0 þ ðqu Þ v h i ðquv Þ þðquv Þ þ 0 þ 0 ¼ 0 Equatin (8) is subtracted frm Eq. (7), and the result is: ðqu Þ v þ ðqu Þ v þ ðqu Þ v ðqu Þ v 5 fflfflfflffl{zfflfflfflffl} fflfflfflffl{zfflfflfflffl} fflfflfflffl{zfflfflfflffl} fflfflfflffl{zfflfflfflffl} 1 þ ðqu Þ v ðqu Þ v þ ðqu Þ v þ ðqu Þ v 5 fflfflfflffl{zfflfflfflffl} fflfflfflffl{zfflfflfflffl} fflfflfflffl{zfflfflfflffl} fflfflfflffl{zfflfflfflffl} ðquv Þ fflfflffl{zfflfflffl} 9 ðquv Þ fflfflffl{zfflfflffl} 10 þ ðquv Þ fflfflffl{zfflfflffl} 11 Þ þ ðquv Þ fflfflffl{zfflfflffl} 1 5 ¼ 0 There are different frms satisfying the abve equatin fr the calculatin f the fluctuating flux. Fr examle, if the ttal value f sentences, 5, and 9 is equal t zer, the ttal amunt f the average f these sentences, i.e.,, 6, and 10, will be zer as well. As a result, in rder fr Eq. (9) t be zer, the summatin f the remaining sentences must be zer. Fr instance, it is ssible t equalize the summatin f terms 1 and with sentence 11 and, als, the summatin f sentences 7 and 8 with term 1. After imlementing these assumtins, Eqs. (0) () are btained: ðquv Þ ¼ 1 ðqu Þ v þ v ð8þ ð9þ ð0þ

20 57 A. JAVADI ET AL. ðquv Þ ¼ 1 ðqu Þ v þ v ðquv Þ ¼ 1 ðqu Þ v þ v ð1þ ðþ After relacing Eqs. (0) () in Eq. (7), the fllwing equatin is btained: ðquv Þ ¼ 1 ðqu Þ v þ v ðþ In general, we can write: qu i u j ¼ 1 qui qu i u j ¼ 1 qui qu i u j ¼ 1 qui qu i u j ¼ 1 qui uj þ uj uj þ uj uj þ uj uj þ uj In the abve equatins, i and j vary frm 1 t and reresent the x, y, and z directins, resectively. Equatins () (7) invlve fluctuating terms in the KEP methd fr three-dimensinal viscus flw. Equatin (16) can be deduced: ¼ 1 þ 1 By searating the mean term and velcity fluctuatins, similar t the rcess described abve, the fllwing equatins are attained t calculate the viscus flux f the mmentum equatin: ¼ 1 ¼ 1 þ 1 þ 1 ðþ ð5þ ð6þ ð7þ ð8þ ð9þ ð50þ APPENDIX B. APPLICATION OF THE KEP METHOD IN THE k AND ɛ EQUATIONS OF THE k ɛ MODEL The subscrits used in this sectin are defined as fllws. The subscrit refers t the main int, subscrits i and j refer t the fllwing ints in the i and j directins, resectively. Mrever, the subscrits m i and m j are reresentative f the earlier ints in the i and j directins, resectively. In rder t clarify the matter, Figure 16 is resented. In this sectin, the discretizatin methd f the k equatin is resented.

21 MODIFICATION OF k Ɛ TURBULENT MODEL 57 Discretizatin f the k Equatin The cnservative frm f the Navier-Stkes equatin is given in (51): q qui qt þ q ð ui u j Þ ¼ q qx i þ m q qu i The discretizatin f the abve equatin is btained as q ui;nþ1 Dt u i;n ð51þ u i þ u i þ q j u j þ u j j u i m þ u i Dx j j u j m þ u j j 5 Dx j " # ð5þ ð Þ jþðqu i = Þ ð Dx j qui = Þ m jþðqu i = Þ Dx j ¼ i m i Dx i þ = The averaged frm f the Navier-Stkes equatin is given in (5): q qui qt þ Figure 16. Index ntatin fr meshes.! qui quj þ qu i quj ¼ q qx i þ m q qu i ð5þ The Navier-Stkes equatins are multilied by u i and, afterwards, their average is written as qu i qui qt þ qui u j qui q ¼ui qxj qx i þ q qu i mui ð5þ

22 57 A. JAVADI ET AL. Equatin (5) is written as a cnservative frm: q qui qt þ q q ð ui u j Þ þ q q ui u j ¼ q qx i þ m q qu i The abve equatin can be discretized as fllws:! q ui;nþ1 u i;n u i þ u i þ q j u j þ u j j u i m þ u i Dt Dx j j u j m þ u j j 5 Dx j u i þ þ q j ui u j þ u j j u i m þ Dx j j ui u j 6 m þ u j j 7 Dx j 5 ¼ i m i Dx i " # ðqu i = Þ j þ ðqu i = Þ ð þ m Dx j qui = Þ m j þ ðqu i = Þ Dx j Equatin (5) is written as a cnservative frm: qu i qui qt þ q ð qui ui u j Þ ¼u i q qx i þ q qu i mui The abve equatin can be discretized as fllws:!! q ui;nþ1 u i;n u i;nþ1 þ u i;n Dt 6 þ q u i þ u i j u j þ u j j Dx j ¼ i m i Dx i u i þ m ð qui = ui u i mj þ ui u j m þ u j j Dx j Þ jþðqu i = Þ Dx j ð qui = Equatin (56) is multilied by u i :!! q ui;nþ1 u i;n u i;nþ1 þ u i;n Dt u i þ u i þ q j u j þ u j j ui Dx j u i m þ u i j u j m þ u j j Dx j u i þ þ q j ui u j þ u j i j u Dx j u i m þ j ui u j 6 m þ u j j Dx j u i u i u i 7 5 Þ m jþðqu i = Dx j ¼ i " # m i Dx i u i þ m ðqu i = Þ j þ ðqu i = Þ Dx j u i ð qui = Þ m j þ ðqu i = Þ Dx j u i Þ u i ð55þ ð56þ ð57þ ð58þ ð59þ

23 MODIFICATION OF k Ɛ TURBULENT MODEL 575 Equatin (59) is subtracted frm Eq. (58): u i;n u i;nþ1 þ u i;n Dt u i þ þ q j ui u i Dx j uj þ u j j ui m þ j ui 6 u i Dx j uj m þ u j 7 j 5 q ui;nþ1 u j þ u j þ q j u j u i Dx j ui þ u i m þ u j 6 j j u i Dx j ui m þ u i 7 j 5 u i þ þ q j ui u j þ u j i j u u i Dx j m þ j ui u j 6 m þ u j j u i 7 Dx j 5 ¼ " i m i u i Dx þ m qu i = þ qu i =qx j j u i i Dx j qui = þ qu i # =qx m j j u i Dx j The secnd term n the left-hand side f Eq. (60) is the cnvective term f the k equatin that is the discretized frm f qu j u i ¼ 1 quj qðu i Þ.Cnsidering the amunt f 1/ entered in the abve equatin, and cnsidering the fllwing arximatins, u i jui ¼ ui þ j ui u i m jui ¼ ui m j þ ui And, als, using the relatin (6) t define the arameter k, k ¼ 1 ui qui The cnvective term f the k equatin is attained as ð60þ ð61þ ð6þ ð6þ q k j þ k 8Dx j u j þ u j j k þ k m j 8Dx j u j m þ u j j ð6þ In rder t calculate the cnvective flux term f the k equatin assing thrugh the interface f ndes and, the fllwing equatin can be written: qu i k ¼ 1 qui k þ k ð65þ

24 576 A. JAVADI ET AL. The last term f Eq. (60) is btained as " m qui = þ qu i =qx j j u i Dx j qui = þ qu i # =qx m j j u i Dx j " ¼ m qui ui = þ qu i j ui = Dx j qui ui = þ qu i # m j ui = Dx j " m qui = þ qu i =qx j j qu i Dx j qui = þ qu i =qx m j j qu i # Dx j ð66þ The secnd art f the right-hand side f the abve equatin is the surce term f the k equatin. T alying the arximatins f (61) and (6), the diffusin term f the k equatin is gained in the frm m ðqk= Þ jþðqk= Dx j Þ ð qk=qxj Þ m jþðqk= Dx j In rder t calculate the diffusin flux term f the k equatin assing thrugh the interface f ndes and, Eq. (68) is written: m qk ¼ m qk þ 1 m qk Imlementing the KEP Methd in the ε Equatin f k ε Mdel In this sectin, the discretizatin methd f the ε equatin is resented. The ε equatin is defined by Eqs. (1). The cnvective term f the ε equatin is discretized in the fllwing frm: 0 1 qn uj þ u j A ui! k ui m u i k! k ui m k Dx j Dx k Dx k þ ui! k ui m k 5 Dx k j! qn uj m þ u j j u i! " k ui m u i k! k ui m k Dx j Dx k Dx k þ ui! # k ui m k Dx k By defining ε as fllws, Þ! m j ð67þ ð68þ ð69þ e ¼ n ui! k ui m k Dx k ð70þ

25 MODIFICATION OF k Ɛ TURBULENT MODEL 577 And cnsidering the arximatins (71) and (7), u i k ui m k Dx k u i k ui m k Dx k!! u i! k ui m k Dx k u i! k ui m k Dx k Eq. (69) is as btained as q e j þ e u j 8Dx þ u j j j j ¼ m j ¼ u i k ui m k Dx k þ ui k ui m k Dx k u i k ui m k þ ui k ui m k Dx k Dx k m j e þ e m j 8Dx j u j m þ u j j In rder t calculate the cnvective flux term f the ε equatin assing thrugh the interface f ndes and, the fllwing equatin can be written: qu i e ¼ 1 qui e þ e ð7þ The diffusin term f the ε equatin is discretized in the frm! mn q u i u i u i k mk! k ui m k Dx j Dx k Dx k þ ui! k ui m u i k! k ui m k 5 Dx k Dx k j "! mn q u i u i u i k mk! k ui m k Dx j Dx k Dx k þ ui! k ui m u i k! # k ui m k Dx k Dx k Cnsidering the arximatins (71) and (7), Eq. (75) is btained as! ðqe= Þ jþðqe= Þ ð m Dx j qe=qxj Þ m jþðqe= Þ Dx j In rder t calculate the diffusin flux term f the ε equatin assing thrugh the interface f ndes and, Eq. (77) is written: m qe ¼ m qe þ 1 m qe j m j ð71þ ð7þ ð7þ ð75þ ð76þ ð77þ